Simplify ( square root of -2)^6

${(\sqrt{- 2})}^{6}$

Step 1

Rewrite $- 2$ as $- 1 (2)$ .

${\sqrt{- 1 (2)}}^{6}$

Step 2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

${(\sqrt{- 1} \cdot \sqrt{2})}^{6}$

Step 3

Rewrite $\sqrt{- 1}$ as $i$ .

${(i \cdot \sqrt{2})}^{6}$

Step 4

Apply the product rule to $i \sqrt{2}$ .

$i^{6} {\sqrt{2}}^{6}$

Step 5

Factor out $i^{4}$ .

$i^{4} i^{2} {\sqrt{2}}^{6}$

Step 6

Rewrite $i^{4}$ as $1$ .

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

${(i^{2})}^{2} i^{2} {\sqrt{2}}^{6}$

Rewrite $i^{2}$ as $- 1$ .

${(- 1)}^{2} i^{2} {\sqrt{2}}^{6}$

Raise $- 1$ to the power of $2$ .

$1 i^{2} {\sqrt{2}}^{6}$

Step 7

Multiply $i^{2}$ by $1$ .

$i^{2} {\sqrt{2}}^{6}$

Step 8

Rewrite $i^{2}$ as $- 1$ .

$- 1 {\sqrt{2}}^{6}$

Step 9

Rewrite ${\sqrt{2}}^{6}$ as $2^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$- 1 {(2^{\frac{1}{2}})}^{6}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 1 \cdot 2^{\frac{1}{2} \cdot 6}$

Combine $\frac{1}{2}$ and $6$ .

$- 1 \cdot 2^{\frac{6}{2}}$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

$- 1 \cdot 2^{\frac{2 \cdot 3}{2}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$- 1 \cdot 2^{\frac{2 \cdot 3}{2 (1)}}$

Cancel the common factor.

$- 1 \cdot 2^{\frac{2 \cdot 3}{2 \cdot 1}}$

Rewrite the expression.

$- 1 \cdot 2^{\frac{3}{1}}$

Divide $3$ by $1$ .

$- 1 \cdot 2^{3}$

Step 10

Simplify the expression.

Raise $2$ to the power of $3$ .

$- 1 \cdot 8$

Multiply $- 1$ by $8$ .

$- 8$

Simplify ( square root of -2)^6

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